A globally convergent BFGS method for nonlinear monotone equations without any merit functions

• Published in: Mathematics of computation 2008-10, Vol.77 (264), p.2231-2240
• Main Authors: Zhou, Wei-Jun, Li, Dong-Hui
• Format: Article
• Language: English
• Subjects: Algorithms; Exact sciences and technology; Global analysis, analysis on manifolds; Hyperplanes; Mathematical functions; Mathematical monotonicity; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Newtons method; Nonlinear equations; Numerical analysis; Numerical analysis. Scientific computation; Perceptron convergence procedure; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-08-02121-2

Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.